It's true that they can be seen starting towards the car without their ticket. Maybe they were called back to get it and that part was edited out. Production usually doesn't help, but in this case there wouldn't be much point to letting them continue towards LAX.(A bonus from examining the video: the only instance of starting-line hustle from Phil as he sprints towards the first elimination mat)

The secret green room was a room they went to at the studio I think? .. it was just a little room they went to when they first arrived in Japan, it had no windows, no furniture it was like a conference room, like a basement that they all slept in and the worst part was they didn't know when they were coming to get them and it was the worst nights sleep they had during the race.